Midterm 1 theory

Question 1

1. Write Euler’s momentum conservation equation for an ideal fluid (vector form) and its z-component using u, v, w.

Answer 1

Vector (Euler) form: ρ(∂u/∂t + (u·∇)u) = −∇p + ρ f
z-component: ρ(∂w/∂t + u ∂w/∂x + v ∂w/∂y + w ∂w/∂z) = −∂p/∂z + ρ f_z

Question 2

2. Explain how incompressible & irrotational flow reduce velocity determination to Laplace’s equation ∇²φ = 0. What are the boundary conditions? When can lift be found from Laplace’s equation?

Answer 2

If flow is irrotational u = ∇φ. If incompressible ∇·u = ∇·(∇φ) = ∇²φ = 0 → Laplace’s equation.
BCs: (a) No-penetration on the body: ∂φ/∂n = V_n (body surface), (b) matching uniform flow at infinity: φ → U·x as |x|→∞.
Lift:
1. Find velocity from solving laplace eq
2. find pressure from Bernoulli using φ,
3. Integrate the pressure or use Kutta-Joukowski if circulation is known

Valid when viscous effects are confined to thin attached boundary layers (high Re) and compressibility is small or treated with compressible potential theory.

Question 3

3. In 2D inviscid incompressible flow over an airfoil with uniform velocity at infinity: what fixes circulation and what’s the physical mechanism? Does it apply to moderately compressible flows?

Answer 3

The Kutta condition fixes circulation. Kutta states that the flow cannot turn around a sharp edge. Therefore, the circulation needs to adjust so that the stagnation point coincides with the trailing edge. This ensures smooth flow off the traling edge and no separation.

The mechanism is viscosity, which forces the stagnation point to move near the trailing edge.

This condition deos not strictly hold in moderately compressible flows but is approximately valid for low subsonic speeds (M<0.3)

Question 4

4. For a 2D airfoil in uniform flow U (incompressible, inviscid, high Re, no separation): relate net aerodynamic force to circulation. What does it predict about drag? Is there approximate truth?

Answer 4

Kutta–Joukowski (per unit span): L’ = ρ U∞ Γ (lift perpendicular to free-stream). Vector: F’ = ρ (U∞ × Γ k̂).
Inviscid potential flow gives zero drag (d’Alembert’s paradox). Approx truth: real flows have nonzero drag from skin friction and separation; for streamlined, attached flows pressure drag can be small, so potential-flow lift predictions are useful but drag must include viscous effects.